If I want to define the wedge product $\vec{u}_1 \wedge \vec{u}_2 \wedge \cdots \wedge \vec{u}_{k}$ of $k$ vectors in $\mathbb{R}^n$, where $2\le k\le n$, as a multilinear form, would the definition: $\left(\vec{u}_{1} \wedge \vec{u}_{2} \wedge \cdots \wedge \vec{u}_{k}\right)\left(\vec{x}_{1}, \vec{x}_{2}, \cdots, \vec{x}_{k}\right) = \mathrm{det}\left(\left[\vec{u}_{i}\bullet\vec{x}_{j}\right]_{1\leqslant i,j \leqslant k}\right)$ be appropriate? Also, as a geometric object, I interpret this wedge product to be an oriented $k$-dimensional oriented paralellepiped which has the ability to act as a function which, to every $k$-tuple of vectors, associates the signed $k$-dimensional volume of the $k$-dimensional paralellepiped spanned by the projections of the $k$-tuple onto the original $k$-dimensional paralellepiped spanned by the vectors in that wedge product.